One-equation Model Research Articles

Partially Lagging One-Equation Turbulence Model

A partially lagging one-equation eddy viscosity model, based on the transformation of the turbulence closure through Bradshaw et al. (“Calculation of Boundary Layer Development Using the Turbulent Energy Equation,” Journal of Fluid Mechanics, Vol. 23, No. 3, 1967, pp. 593–616), is proposed. The model attempts to account for the turbulence history effects by spatially lagging the destruction/diffusion terms. The amount of relaxation is based on the von Kármán length scale. The present model does not assume equal diffusion coefficients of the and equations; therefore, third-order velocity gradients emerge. The performance of the present model is assessed against the Spalart–Allmaras model (“A One-Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper 1992-0439, 1992) and the one-equation model proposed by Menter (“Eddy Viscosity Transport Equations and Their Relation to the Model,” Journal of Fluids Engineering, Vol. 119, 1997, pp. 876–884), which is based on Bradshaw et al. (“Calculation of Boundary Layer Development Using the Turbulent Energy Equation,” Journal of Fluid Mechanics, Vol. 23, No. 3, 1967, pp. 593–616), and the assumption of equal diffusion coefficients. The behavior of the one-equation transformed model without the presence of third-order velocity gradients is also examined. The proposed model is validated through a comparison to a wide class of internal and external turbulent flows.

Diffusive Transport in Naturally Fractured Reservoirs: Averaging Equations

In this article a mass transfer model for multiphase flow in naturally fractured reservoirs was derived. The scaling analysis is presented in order to identify the characteristic scales where the fractured porous media can be studied. The system under consideration consists of two regions and an inter-region. One region is porous medium saturated with multiphase flow (-region) and the other region is multiphase flow in the fracture (-region). The multiphase flow analyzed in both regions is water-oil-gas. The one-equation model for the fractured system was obtained considering thermodynamic equilibrium, which can be applied in -region, -region, and -inter-region due to the non-local averaging approximation.

Numerical simulation of airflow and temperature fields around an occupant in indoor environment

An accurate prediction of flow and thermal fields around an human body provides valuable information in designing an efficient ventilation system. This study tries to address this issue by investigating the flow structure around an human body subjected to a displacement ventilation system through the computational fluid dynamics (CFD). For this purpose, both the large eddy simulation (LES) and hybrid LES–RANS methods are employed. The RAST one-equation model (OEM) and the dynamic Smagorinsky model (DSM) are utilized as a sub-grid scale (SGS) modeling for the LES while for the hybrid LES–RANS method, the SST–SAS version is applied. Predicted results are compared with available experimental measurements in the literature. Comparisons show that the OEM has a better performance in reproducing the correct level of velocity and temperature fields around the body as well as in other locations of the room while the SST–SAS model fails to predict these quantities accurately, especially around the occupant's body. Above all, the OEM provides a good compromise between accuracy and robustness in predicting the airflow and temperature fields in an indoor environment.

Analytical Study of Heat Flux Splitting in Micro-channels Filled with Porous Media

The conditions of occurring heat flux splitting (bifurcation) phenomenon in a micro-channel filled with a porous medium including internal heat generations within both the solid and fluid phases under the local thermal non-equilibrium (LTNE) condition is analytically studied in the slip regime. The channel walls are subjected to a constant heat flux. Exact solutions for both the dimensionless temperatures of the two phases and the Nusselt number are obtained. Effects of the pertinent parameters such as heat generation parameter (\(\omega \)), the interphase heat transfer parameter (Vadasz in J Porous Media 15:249–258, 2012) or Biot number (\(\textit{Bi}\)), the fluid-to-solid effective conductivity ratio (k), and the temperature jump coefficient (\(\beta \)) on the dimensionless temperature profile (\(\theta \)) of the two phases as well as the Nusselt number are investigated. Moreover, the validity of one-equation model (the local thermal equilibrium assumption) is analyzed by comparing the Nusselt number obtained by one-equation model (LTE) with that obtained by the two-equation model (LTNE). Results reveal that the conditions at which the heat flux bifurcation (splitting) occurs in the slip regime is the same as those of the no-slip regime. In addition, a kind of heat flux bifurcation in which the solid and fluid phases have the same dimensionless temperature sign is observed in the slip regime, while it was not previously observe in the no-slip regime. It is discussed that the Nusselt number can increase or decrease with respect to \(\omega \) and may have either positive or negative values in both the no-slip and slip regimes. The presence of internal heat generation intensifies the role of \(\beta \) in the Nusselt number reduction. In addition, the accuracy of LTE model increases with increased \(\textit{Bi}\) and with decreased \(\beta \), while it is not a monotonic function of k in the presence of internal heat generation.

Low-Reynolds-Number One-Equation Turbulence Model Based on Closure

Accurate turbulence modeling remains a critical problem in the prediction capability of computational fluid dynamics. In this paper, a new one-equation eddy-viscosity model is derived from closure. The equation used in this derivation includes a cross-diffusion term that allows the new model to switch between the model properties exhibited by the two-equation or turbulence models. In addition, the damping function used in conjunction with the proposed one-equation model has not been studied before in the literature. The new model is used to simulate several benchmark canonical flows involving both the free shear layer and wall-bounded turbulent flows with small separation regions. The open-source software OpenFOAM is used for the flowfield calculations. It is shown that the new model improves the accuracy of flow simulations compared to the widely used one-equation Spalart–Allmaras model and is competitive with the shear-stress-transport model.

Multiphase Flow in a Naturally Fractured Reservoir: Equation of Continuity

In this article an equation of continuity of multiphase flow in naturally fractured reservoirs was derived. The scaling analysis is presented in order to identify the characteristic scales where the fractured porous media can be studied. The system under consideration consists of two regions and an inter-region. One region is porous medium saturated with multiphase flow (-region) and the other region is multiphase flow in the fracture (-region). The multiphase flow analyzed in both regions is water-oil-gas. The one-equation model (single velocity) was obtained for predicted energy transfer in naturally fractured reservoirs using the non-local averaging approximation.

Development and Validation of a Fluid–Structure Solver for Transonic Panel Flutter

A partitioned fluid–structure coupling code for transonic panel flutter has been developed and validated. The Reynolds-averaged Navier–Stokes equations are solved numerically by means of an implicit finite volume method to account for nonlinear aerodynamics, as there are shock waves and a viscous boundary layer at the panel surface. An implicit finite element formulation of the structural equations, as well as a Galerkin solution of the von Kármán plate equation are employed to solve elastic panel deformations with respect to geometric nonlinearities. A detailed validation process has shown good agreement with results from the literature for high subsonic and low supersonic Mach numbers. Thereby, a recent comparison between theory and experiment is confirmed. The validated solver is then used for further studies focusing on the impact of turbulent boundary layers on aeroelastic stability boundaries and instabilities. An increase in aerodynamic damping due to a viscous boundary layer is identified by an increase of the aeroelastic stability boundary. Furthermore, a significant damping on high flutter frequencies and mode shapes is revealed. The application of either a one-equation or two-equation turbulence model did not cause any major deviations in the results.

Assessing the business values of information technology and e-commerce independently and jointly

The purpose of this paper is to assess the business values of information technology (IT) and electronic commerce (EC) independently and simultaneously as measured by productive efficiency, in order to provide new insights into IT and EC investments and, consequently, lead to better decisions on IT and EC investments. The paper analyzes a panel data set at the country level based on the theory of production and its companions called the time-varying stochastic production frontier (SPF) approaches with the one-equation and two-equation models estimated by a two-step nonlinear maximum-likelihood method. The performance metric called productive efficiency is built in the research approaches. The empirical evidence strongly suggests that the presence of EC may strengthen or weaken IT value, and vice versa, which provide a good explanation for the disappearance or existence of the so-called productivity paradox, and that the paradox may exist in a country regardless of whether it is a developed or a developing country, inconsistent with conventional wisdom claiming that the paradox exists only in developing countries. The findings imply that it is imperative to carefully assess the values of IT and EC and, as such, prudent rather than blind IT and EC investing decisions are made, and that the values of IT and EC must be evaluated jointly rather than separately. The findings add significant contributions to the literature and serve as a catalytic agent in stimulating further comparative research in these important areas linking IT investments and EC developments when their business values are the major concern.

Multiphase Flow in a Naturally Fractured Reservoir: Heat Transfer

In this article, an energy transport model of multiphase flow in a naturally fractured reservoir was derived. The scaling analysis is presented in order to identify the characteristic scales where the fractured porous media can be studied. The system under consideration consists of two regions and an inter-region. One region is porous medium saturated with multiphase flow (η-region) and the other region is multiphase flow in the fracture (ω-region). The multiphase flow analyzed in both regions is water-oil-gas. The one-equation model (single temperature) was obtained for predicting energy transfer in a naturally fractured reservoir using the non-local averaging approximation.

Large-eddy simulation of shear flows and high-speed vaporizing liquid fuel sprays

Large eddy simulation models are tested for use on high speed evaporating liquid sprays. Three models are tested: standard Smagorinsky, a one-equation viscosity based model, and the dynamic structure model. The models are tested using channel flow, planar gas jet, and a diesel spray. The motivation of the work is to evaluate the accuracy and suitability of LES models that can be used for internal combustion engine modeling with direct injection of fuel in which moderate mesh resolution is the norm. Results from the channel flow and the gas jet provide additional insight into model capabilities that impact performance in liquid spray simulations. Simulation results were analyzed using contour images to compare the general structure of the flows, and experimental data for comparison of ensemble averaged mean and rms velocity profiles, liquid and vapor penetration, and mass fraction profiles. Results show that all three models perform similarly in the channel flow with good matches on the mean velocity results and some under prediction of rms values. The dynamic structure model showed slightly steeper near wall rms values closer to the experimental data. In the jet flow, the one-equation model results showed unexpected flow structures in images of the velocity magnitude. Mean velocity profiles matched data well for all three models. But centerline velocity decay rates and rms velocity radial profiles matched data much better for the dynamic structure model results. In the liquid spray the one-equation model performed poorly when compared to experimental data. The Smagorinsky model gave reasonable results and the dynamic structure model gave very good results in these comparisons. The overall conclusion is that the dynamic structure model is the best of the three models for direct injection engine simulations.

Numerical Simulations of the Sandia Flame D Using the Eddy Dissipation Concept

A turbulent piloted methane/air diffusion flame (Sandia Flame D) is calculated using both compressible Reynolds-averaged and large-eddy simulations (RAS and LES, respectively). The Eddy Dissipation Concept (EDC) is used for the turbulence-chemistry interaction, which assumes that molecular mixing and the subsequent combustion occur in the fine structures (smaller dissipative eddies, which are close to the Kolmogorov length scales). Assuming the full turbulence energy cascade, the characteristic length and velocity scales of the fine structures are evaluated using a standard k- 𝜖 turbulence model for RAS and a one-equation eddy-viscosity sub-grid scale model for LES. Finite-rate chemical kinetics are taken into account by treating the fine structures as constant pressure and adiabatic homogeneous reactors (calculated as a system of ordinary-differential equations (ODEs)) described by a Perfectly Stirred Reactor (PSR) concept. A robust implicit Runge-Kutta method (RADAU5) is used for integrating stiff ODEs to evaluate reaction rates. The radiation heat transfer is treated by the P1-approximation. The assumed β-PDF approach is applied to assess the influence of modeling of the turbulence-chemistry interaction. Numerical results are compared with available experimental data. In general, there is good agreement between present simulations and measurements both for RAS and LES, which gives a good indication on the adequacy and accuracy of the method and its further application for turbulent combustion simulations.

Analysis of Empirical Constant of Eddy Viscosity by Zero- and One-Equation Turbulence Model in Wake Simulation

In this paper, the wakes behind a square cylinder were simulated using two kinds of different turbulence models for the eddy viscosity concept such as the zero- and the one-equation model in which the former is the mixing length model and the latter is the k-equation model. For comparison between numerical and analytical solutions, we employed three skill assessments: the correlation coefficient(r) for the similarity of the wake shape, the error of maximum velocity difference(EMVD) for the accuracy of wake velocity and the ratio of drag coefficient(RDC) for the pressure distribution around the structure. On the basis of the numerical results, the feasibility of each model for wake simulation was discussed and a suitable value for the empirical constant was suggested in these turbulence models. The zero-equation model, known as the simplest turbulence model, overestimated the EMVD and its absolute mean error(AME) for r , EMVD and RDC was ranging from 20.3 % to 56.3 % for all test. But the AME by the one-equation model was ranging from 3.4 % to 19.9 %. The predicted values of the one-equation model substantially agreed with the analytical solutions at the empirical mixing length scale L+0.6b 1/2 with the AME of 3.4 %. Therefore it was concluded that the one-equation model was suitable for the wake simulation behind a square cylinder when the empirical constant for eddy viscosity would be properly chosen.

The Investigation of Turbulence Models Based on Numerical Simulation

The algebraic model, one-equation model, two equation models are Analyzed in the paper. Focusing on the k-ε turbulent model contains (the standard k-εturbulence model, RNG k-εmodel, Realizable k-ε model), and the origin of the shear stress transport model, Reynolds stress equation model on mathematical and physical model is discussed.the advantages and disadvantages of each model is Pointed out ,And the future development direction of turbulent model and improvement measures are proposed on the paper.

Simulation of flow in compound open-channel using a discontinuous Galerkin finite-element method with Smagorinsky turbulence closure

The small-scale spatial variability of eddy viscosity which is characteristic for the turbulent shear stress in compound open-channel flows was studied and investigated in this paper. Different options including a constant value, zero-equation, one-equation, two-equation, and Smagorinsky turbulence models for parameterizing the eddy viscosity were developed in the framework of the discontinuous Galerkin finite-element SLIM model and applied for presenting the complex velocity profile in two different experimental data sets of laboratory flumes. A very good qualitative agreement was achieved between numerical results and measurement data for both velocity and flow depth of all experimental data sets in general. In addition, the calculation results showed that the turbulent Smagorinsky empiricism allowed a better presentation of non-uniform velocity in the floodplain and transition regions between plain and main channels than the others in all calculated cases. This empiricism predicted a very close variation of eddy viscosity in comparison with the results calculated by the depth-averaged Reynolds' stress and the lateral gradient of longitudinal velocity. The eddy viscosity varies significantly in the channel section; in particular the small values often occurred around the middle location of floodplains and the central location of the main channel while the large values appeared in the transition regions, presenting different minimum and maximum values of eddy viscosity in each flow region. The effects of eddy viscosity variation on lateral distribution of velocity profile were also investigated and discussed.

Large eddy simulations of the influence of piston position on the swirling flow in a model two-stroke diesel engine

Purpose – The purpose of this paper is to study the effect of piston position on the in-cylinder swirling flow in a simplified model of a large two-stroke marine diesel engine. Design/methodology/approach – Large eddy simulations with four different models for the turbulent flow are used: a one-equation model, a dynamic one-equation model, a localized dynamic one-equation model and a mixed-scale model. Simulations are carried out for two different geometries corresponding to 100 and 50 percent open scavenge ports. Findings – It is found that the mean tangential profile inside the cylinder changes qualitatively with port closure from a Lamb-Oseen vortex profile to a solid body rotation, while the axial velocity changes from a wake-like profile to a jet-like profile. The numerical results are compared with particle image velocimetry measurements, and in general, the authors find a good agreement. Research limitations/implications – Considering the complexity of the real engine, the authors designed the engine model using the simplest configuration possible. The setup contains no moving parts, the combustion is neglected and the exhaust valve is discarded. Originality/value – Studying the flow in a simplified engine model, the setup allows studies of fundamental aspects of swirling flow in a uniform scavenged engine. Comparing the four turbulence models, the local dynamic one-equation model is found to give the best agreement with the experimental results.

A numerical investigation into the aerodynamic effect of pressure pulses on a tunnel ventilation fan

Tunnel ventilation fans are subjected to pressure pulses as a consequence of trains passing the ventilation shafts within which they are installed. These pressure pulses alter the volume flow rate through a ventilation fan, and consequently the static pressure field around fan blades. Today’s trains typically travel through railway tunnels faster than has been the historic norm. Additionally, platform screen doors are now a standard feature of metro systems. Both result in the trains involved inducing larger pressure pulses than industrial fan designers have traditionally assumed. Although tunnel ventilation fan in-service failures are rare, engineers increasingly associate those failures with new or renovated tunnel systems with larger pressure pulses. Consequently, we require insight into the aerodynamic and mechanical consequences that occur with subjecting a tunnel ventilation fan to a pressure pulse. In the research reported in this paper, the authors model the pressure pulse induced by a train in a tunnel system as a rapidly changing fan volume flow rate. The authors computed the fan operating point by means of a large eddy simulation with a one-equation sub-grid scale turbulence model. The authors undertook the computation using the open source computational fluid dynamic code OpenFOAM. An analysis of the computational results provides an insight into the effects of turbulent structures that develop within the fan blade passage. The analysis indicates that a pressure pulse too brief to drive a tunnel ventilation fan into stall still results a doubling of the unsteady aerodynamic and mechanical forces to which the blade is subjected. A doubling of unsteady mechanical forces as a consequence of a pressure pulse constitutes an increase significant enough to contribute to the tunnel ventilation fan’s in-service mechanical failure.

Tangential synthetic jets for separation control

A numerical study of separation control has been made to investigate aerodynamic characteristics of a NACA23012 airfoil with a tangential synthetic jet. Simulations are carried out at the chord Reynolds number of Re=2.19×106. The present approach relies on solving the Unsteady Reynolds-Averaged Navier–Stokes (URANS) equations. The turbulence model used in the present computation is the Spalart–Allmaras one-equation model. All computations are performed with a finite volume based code. Stall characteristics are significantly improved by controlling the formation of separation vortices in the flow. We placed the synthetic jet at the 12% chord, xj=0.12c, where we expected the separation to occur. Two distinct jet oscillating frequencies: Fj+=0.159 and Fj+=1 were considered. We studied the effect of blowing ratio, Vj/U∞, where it was varied from 0 to 5. The inclined angle of the synthetic jet was varied from αj=0° up to αj=83°. For the non-zero inclined angles, the local maximum in the aerodynamic performance, Cl/Cd, of 6.89 was found for the inclined angle of about 43°. In the present method, by means of creating a dent on the airfoil, linear momentum is transferred to the flow system in tangential direction to the airfoil surface. Thus the absolute maximum of 11.19 was found for the tangential synthetic jet at the inclined angle of the jet of 0°. The mechanisms involved for a tangential jet appear to behave linearly, as by multiplying the activation frequency of the jet by a factor produces the same multiplication factor in the resulting frequency in the flow. However, the mechanisms involved in the non-zero inclined angle cases behave nonlinearly when the activation frequency is multiplied.

Nonequilibrium energy spectrum in the subgrid-scale one-equation model in large-eddy simulation

The subgrid-scale (SGS) modeling in large-eddy simulation (LES) which accounts for the effect of unsteadiness and nonequilibrium state in the SGS is considered. Unsteadiness is incorporated by considering the spectral evolution in the forced homogeneous isotropic turbulence using the transport equation for the SGS energy. As for the unfiltered spectrum, perturbative expansion of the Kovasnay spectral model about the Kolmogorov −5/3 energy spectrum which constitutes a base equilibrium state in the inertial subrange, yields the extra components with −7/3 and −9/3 powers. It is shown that these spectra are actually extracted in the direct numerical simulation (DNS) data and these components govern the unsteady energy transfer. As for the SGS real-space representation of the spectral model, we consider the SGS one-equation model. The perturbation expansion is applied to the one-equation model by setting the base SGS energy as the standard Smagorinsky model, which assumes the equilibrium state in the SGS and its spectral counterpart is the Kolmogorov −5/3 spectrum. The solution yields the terms whose spectral counterparts are the components with −7/3 and −9/3 powers. These additional terms are induced by temporal variations of the base SGS energy. In the temporal variations of the grid-scale energy, SGS energy, SGS production term, and SGS dissipation which are obtained by applying the filter to the DNS data, it is shown that these quantities lag in time in this order. This time-lag is not realized in the standard Smagorinsky model and the one-equation model because the SGS dissipation is defined so that it instantaneously adjusts to the SGS energy. In the one-equation model, the direction of the energy cascade in the initial period is opposite to that obtained in the DNS data. To retrieve correct time-lag and direction of energy transfer, we relax this instantaneous adjustment and propose the nonequilibrium Smagorinsky model. In this nonequilibrium model, the SGS energy incurred by the −7/3 spectrum is added to the base Smagorinsky energy. Assessment in actual LES shows that the time-lag predicted using the standard Smagorinsky and the one-equation models is inaccurate, whereas good agreement with the DNS data is achieved in the nonequilibrium Smagorinsky model. Extraction of the grid-scale nonequilibrium energy spectrum yields the −7/3 and −9/3 components in addition to the base −5/3 spectrum. In the nonequilibrium Smagorinsky model, continuation of the grid-scale spectra into the SGS is established for the −5/3 and −7/3 components. As a result, the unsteady energy transfer is more accurately predicted, whereas the standard Smagorinsky model does not have the SGS counterpart for the −7/3 component. Feasibility of employing the eddy-viscosity approximation to account for the transfer in the period in which −9/3 spectrum prevails is discussed.

Effect of turbulence models on predicting HAWT rotor blade performances

The effect of turbulence models on predicting the aerodynamic performance of horizontal-axis wind turbine (HAWT) rotor blades has been investigated. The flow fields around the 2-D airfoil and the 3-D blade of the NREL phase VI wind turbine rotor have been analyzed, and the results are compared between a correlation-based transition model and two other fully turbulent models. The turbulence models selected are the Spalart-Allmaras fully turbulent one-equation model, the k-ω SST fully turbulent two-equation model, and the transition model. A vertex-centered finite-volume method based on an unstructured mesh technique was used to discretize the governing Navier-Stokes equations. The inviscid fluxes were calculated by using 2nd order Roe’s FDS, and the viscous fluxes were evaluated in a central-differencing manner. For the time integration, an implicit method based on the Gauss-Seidel iteration was used. The results showed that the transition model well captures the laminar separation bubbles on the surface of the airfoil and the blade, and these separation bubbles trigger the separation-induced transition as the laminar flow separates and re-attaches as turbulent. The separation bubbles change the flow pattern on the surface of the airfoil and on the blade, and the pressure and skin-friction distributions are also changed abruptly across the laminar-turbulent transition. With properly predicted boundary-layer transition, the results of the transition model match well with the experiment. However, the results of the fully turbulent models deviate from the experiment due to the lack of the ability of capturing the boundary-layer transition. The adoption of a proper transition turbulence model is essential for the accurate prediction of the aerodynamic loads and also the rotor performance for horizontal-axis wind turbines.

Wind-field and pollution-dispersion simulation in a street canyon in Helsinki with ADREA-HF code

ADREA-HF, which is a Computational Fluid Dynamics (CFD) code, is utilised in order to numerically study the flow and concentration fields within a street-canyon area. The selected site is Runeberg Str., a typical urban street canyon with an aspect ratio of approximately 1:1 in Helsinki, Finland. The ADREA-HF model is a transient, non-hydrostatic, dense transport code, especially developed for dispersion modelling of buoyant or passive gases over complex terrain in local scale. It solves the 3D unsteady Reynolds Averaged Navier-Stokes (RANS) equations treating complex multi-building domains with a porosity formulation. For modelling turbulence a one-equation eddy-viscosity model is used. The numerical results illustrate the flow and concentration fields within the canyon and also show the influence of the detailed geometry, such as, that of the street junction situated at the northern end of the canyon, and that of the boulevard at the southern end.